Georg Ferdinand Ludwig Philipp Cantor
1845-1918
German Mathematician
Georg Cantor is regarded as the founder of set theory. He also introduced the concept of transfinite numbers. This is the idea that there is not just one quantity of "infinity," but many quantities that are distinct from one another but all indefinitely large.
Cantor was born in St. Petersburg, Russia, on March 3, 1845. His father was a well-to-do merchant, and his mother was an artistic woman from a family of musicians. In 1856, the family moved to Germany. Cantor's mathematical talents soon became apparent at gymnasien, or secondary schools, in Darmstadt and Wiesbaden.
As an undergraduate student at the University of Berlin, Cantor specialized in mathematics, physics, and philosophy. There, he first met the mathematician Leopold Kronecker (1823-1891), who was later to become his archrival. In 1867, Cantor received his doctorate from the University of Göttingen, with a thesis entitled In re mathematica ars propendi pluris facienda est quam solvendi, or, "In mathematics the art of asking questions is more valuable than solving problems." The thesis addressed an unsettled question from Disquisitiones Arithmeticae, an 1801 work by Carl Friedrich Gauss (1777-1855).
Cantor spent a short time teaching at a girls' school in Berlin, and then joined the faculty at the University of Halle. He remained there for the rest of his life. Between 1869 and 1873, he published a series of papers on number theory and trigonometric series. It was in the 1870s that he began working on set theory, which led him to the concept of transfinite numbers.
A set is a group of objects or numbers that retain their individuality while having some property in common. The group can be either finite or infinite. For example, the set of students in a class is finite. The set of integers is infinite. Cantor concerned himself in particular with one-to-one correspondences. That is, the sets {a,b,c} and {1,2,3} are in a one-to-one correspondence because a can be paired with 1, b can bepaired with 2, and c can be paired with 3. However, for infinite sets, one cannot simply compare the number of members, since the sets are indefinitely large.
Georg Cantor. (The Library of Congress. Reproduced by permission.)Cantor developed a theory of countability based on one-to-one correspondences between infinite sets. Rational numbers, for example, are countable even though infinite. They can be placed in a one-to-one correspondence with integers. The real numbers, consisting of the set of rational and irrational numbers taken together, are uncountable. This led him to the realization that some infinite sets are larger than others, and thus to the idea of transfinite numbers.
Cantor's first paper on set theory was at first refused for publication by Crelle's Journal due to the vehement opposition of Leopold Kronecker. Most scholarly journals ask professionals in the field to referee papers before publishing, and Kronecker was serving in that capacity. The paper was published the next year, but the antagonism between the two mathematicians remained. Its basis was essentially philosophical. Cantor drew on his childhood religious training and Platonic metaphysics to come to the conclusion that infinite numbers had an actual existence. Kronecker was more limited in his view. "God made the integers," he asserted, "and all the rest is the work of man."
As for the work of Cantor, it became the basis for an entire field of study, the mathematics of the infinite. It was fundamental to the development of function theory, analysis, and topology. It also changed educators' ideas about the foundations of mathematical thought. Basic set theory and functions are part of today's elementary mathematics curriculum, first introduced in the 1960s as part of the "new math."
In 1874, Cantor married Vally Guttman. They had five children. The mathematician suffered from bouts of depression from about 1884, but continued to work as his health permitted. In 1897 he was involved in organizing the first international mathematical congress, which took place in Zurich. He died in Halle on January 6, 1918.
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